118 research outputs found
The proof-theoretic strength of Ramsey's theorem for pairs and two colors
Ramsey's theorem for -tuples and -colors () asserts
that every k-coloring of admits an infinite monochromatic
subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and
two colors, namely, the set of its consequences, and show that
is conservative over . This
strengthens the proof of Chong, Slaman and Yang that does not
imply , and shows that is
finitistically reducible, in the sense of Simpson's partial realization of
Hilbert's Program. Moreover, we develop general tools to simplify the proofs of
-conservation theorems.Comment: 32 page
The Parametric Complexity of Lossy Counter Machines
The reachability problem in lossy counter machines is the best-known ACKERMANN-complete problem and has been used to establish most of the ACKERMANN-hardness statements in the literature. This hides however a complexity gap when the number of counters is fixed. We close this gap and prove F_d-completeness for machines with d counters, which provides the first known uncontrived problems complete for the fast-growing complexity classes at levels 3 < d < omega. We develop for this an approach through antichain factorisations of bad sequences and analysing the length of controlled antichains
Foundations of Online Structure Theory II: The Operator Approach
We introduce a framework for online structure theory. Our approach
generalises notions arising independently in several areas of computability
theory and complexity theory. We suggest a unifying approach using operators
where we allow the input to be a countable object of an arbitrary complexity.
We give a new framework which (i) ties online algorithms with computable
analysis, (ii) shows how to use modifications of notions from computable
analysis, such as Weihrauch reducibility, to analyse finite but uniform
combinatorics, (iii) show how to finitize reverse mathematics to suggest a fine
structure of finite analogs of infinite combinatorial problems, and (iv) see
how similar ideas can be amalgamated from areas such as EX-learning, computable
analysis, distributed computing and the like. One of the key ideas is that
online algorithms can be viewed as a sub-area of computable analysis.
Conversely, we also get an enrichment of computable analysis from classical
online algorithms
Open questions about Ramsey-type statements in reverse mathematics
Ramsey's theorem states that for any coloring of the n-element subsets of N
with finitely many colors, there is an infinite set H such that all n-element
subsets of H have the same color. The strength of consequences of Ramsey's
theorem has been extensively studied in reverse mathematics and under various
reducibilities, namely, computable reducibility and uniform reducibility. Our
understanding of the combinatorics of Ramsey's theorem and its consequences has
been greatly improved over the past decades. In this paper, we state some
questions which naturally arose during this study. The inability to answer
those questions reveals some gaps in our understanding of the combinatorics of
Ramsey's theorem.Comment: 15 page
A Formalization of Forcing and the Unprovability of the Continuum Hypothesis
We describe a formalization of forcing using Boolean-valued models in the Lean 3 theorem prover, including the fundamental theorem of forcing and a deep embedding of first-order logic with a Boolean-valued soundness theorem. As an application of our framework, we specialize our construction to the Boolean algebra of regular opens of the Cantor space 2^{omega_2 x omega} and formally verify the failure of the continuum hypothesis in the resulting model
Metastable convergence theorems
The dominated convergence theorem implies that if (f_n) is a sequence of
functions on a probability space taking values in the interval [0,1], and (f_n)
converges pointwise a.e., then the sequence of integrals converges to the
integral of the pointwise limit. Tao has proved a quantitative version of this
theorem: given a uniform bound on the rates of metastable convergence in the
hypothesis, there is a bound on the rate of metastable convergence in the
conclusion that is independent of the sequence (f_n) and the underlying space.
We prove a slight strengthening of Tao's theorem which, moreover, provides an
explicit description of the second bound in terms of the first. Specifically,
we show that when the first bound is given by a continuous functional, the
bound in the conclusion can be computed by a recursion along the tree of
unsecured sequences. We also establish a quantitative version of Egorov's
theorem, and introduce a new mode of convergence related to these notions
A Semantics-Based Approach to Design of Query Languages for Partial Information
Most of work on partial information in databases asks which operations of standard languages, like relational algebra, can still be performed correctly in the presence of nulls. In this paper a different point of view is advocated. We believe that the semantics of partiality must be clearly understood and it should give us new design principles for languages for databases with partial information. There are different sources of partial information, such as missing information and conflicts that occur when different databases are merged. In this paper, we develop a common semantic framework for them which can be applied in a context more general than the flat relational model. This ordered semantics, which is based on ideas used in the semantics of programming languages, cleanly intergrates all kinds of partial information and serves as a tool to establish connections between them. Analyzing properties of semantic domains of types suitable for representing partial information, we come up with operations that are naturally associated with those types, and we organize programming syntax around these operations. We show how the languages that we obtain can be used to ask typical queries about incomplete information in relational databases, and how they can express some previously proposed languages. Finally, we discuss a few related topics such as mixing traditional constraints with partial information and extending semantics and languages to accommodate bags and recursive types
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